2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128978An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space.14 pagesDifferential GeometryComplex Variables53C55; 32Q15; 32T15text